Commit cacef22c authored by Leonard Guetta's avatar Leonard Guetta
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\chapter{Homotopy and homology type of free $2$-categories}
\section{Preliminaries : the case of free $1$-categories}
In this section, we review some homotopical results concerning free ($1$-)categories that will be of great help in the sequel.
A \emph{reflexive graph} $G$ consists of the data of two sets $G_0$ and $G_1$ together with three maps
G_0 \ar[r,"1_{(-)}"] & G_1 \ar[l,"\src",bend right] \ar[l,"\trgt",bend left]
such that $\src \circ \iota = \trgt \circ \iota = \mathrm{id}_{G_0}$. In particular, the map $1_{-}$ is a monomorphism. The same vocabulary as for categories is used : elements of $G_0$ are \emph{objects} or \emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells}, arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A \emph{morphism of reflexive graphs} $ f : G \to G'$ consists of maps $f_0 : G_0 \to G'_0$ and $f_1 : G_1 \to G'_1$ that commute with sources, targets and units in an obvious sense. This defines the category $\Rgrph$ of reflexive graphs. There is a ``underlying reflexive graph'' functor
U : \Cat \to \Rgrph,
which has a left adjoint
L : \Rgrph \to \Cat.
For a reflexive graph $G$, the objects of $L(G)$ are exactly the objects of $G$ and an arrow $f$ of of $L(G)$ is a chain
X_0 \overset{\rightarrow}{f_1} X_1 \overset{\rightarrow}{f_2} X_2 \rightarrow \cdots \rightarrow X_{n-1} \overset{\rightarrow}{f_n} X_{n}
of composable arrows of $G$, such that \emph{none} of the $f_k$ are units. The integer $n$ is referred to as the \emph{length} of $f$. Composition is given by concatenation of chains.
A category $C$ is free in the sense of \todo{ref} if and only if there exists a reflexive graph $G$ such that
C \simeq L(G).
If $C$ is free, consider the reflexive graph $G$ such that $G_0 = C_0$ and $G_1$ is the subset of $C_1$ whose elements are either generating $1$-cells of $C$ or units. It is straightforward to check that $C\simeq L(G)$.
Conversely, if $C \simeq L(G)$ for some reflexive graph $G$, then the description of the arrows of $L(G)$ given in the previous paragraph shows that $C$ is free and that its set of generating $1$-cells is (isomorphic to) the non unital $1$-cells of $G$.
Note that for a morphism of reflexive graphs $ f : G \to G'$, the functor $L(f)$ is not necessarily rigid in the sense of \todo{ref} because generating $1$-cells may be sent to units.
There is another important description of the category $\Rgrph$. Consider $\Delta_{\leq 1}$ the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$. Then, the category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq 1}}$, the category of pre-sheaves on $\Delta_{\leq 1}$.
......@@ -13,6 +13,7 @@
......@@ -91,6 +91,7 @@
\newcommand{\CCat}{\underline{\mathbf{Cat}}} %2-category of small categories
\newcommand{\CCAT}{\underline{\mathbf{CAT}}} %2-category of big categories
......@@ -110,6 +111,10 @@
\def\fcomp_#1{\mathbin{\hat{\underset{#1}{\ast}}}} % formal composition
% source and targets
% useful stuff
\newcommand{\ii}{\mathbf{i}} % a boldfont i
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